This invention relates generally to a method of estimating physical properties of three-dimensional porous media. More particularly, it relates to estimating physical properties of rock, bone, soils, ceramics, granular media, and other composites.
There has been long-term interest in the estimation of physical properties of three-dimensional (3D) porous media using numerical methods. For example, estimation of permeability from rock samples and well log data is of great importance in petroleum exploration. The prediction of permeability is one of the most important challenges in quantitative rock physics. Many relatively successful and common methods are empirical ones, such as the widely known Kozeny-Carman Relation based on simple cylindrical pore geometry (see for example Mavko et al., xe2x80x9cThe Rock Physics Handbook,xe2x80x9d Cambridge University Press, Cambridge, England, 1998, pp. 260-264). This relation provides a way to relate the permeability of porous media to some parameters, like porosity, surface area, grain size, etc.
Walsh and Brace, xe2x80x9cThe Effect of Pressure on Porosity and the Transport Properties of Rock,xe2x80x9d J. of Geophysical Research, V.89, no. B11, Oct. 10, 1984, pp. 9425-9431, related the permeability to porosity, geometrical factor, formation factor, and specific surface area, and provided a good means of correlating the data on permeability and formation factor for low porosity and low permeability granites. Conventional methods for permeability prediction from thin sections use two-point correlation of thin section images (see for example Blair et al., xe2x80x9cTwo-point Correlation Functions to Characterize Microgeometry and Estimate Permeabilities of Synthetic and Natural Sandstones,xe2x80x9d Lawrence Livermore National Laboratory Report UCRL-LR-114182, Livermore, Calif., August 1993). These methods can predict permeability within a reasonable range of errors. However, the specific surface area needs to be determined from the correlation function. This determination requires high resolution and is very sensitive to scale, which can introduce errors in the estimation. In addition, they still need empirical parameters, formation factor and geometrical factor, that are hardly ever measured directly from thin sections.
Blair et al. (1993, cited above) suggested a method for estimating porosity and the specific surface area through image processing of thin sections. They then used the estimated parameters from the empirical relation by Walsh and Brace (1984, cited above) for estimation of permeability, providing a good result in estimating permeability of some sandstones. However, there are typically limitations to estimating the parameters. The formula for estimation of permeability shows strong dependency on the specific surface area, which is difficult to determine and very sensitive to the scale of the image. On the other hand, the formation factor derived from electric properties of rocks and the geometric factor is not directly measurable from the image. Empirical estimates for these parameters are still needed.
There is a need, therefore, for a methodology and a system to estimate numerically physical properties of three-dimensional (3D) porous media with high accuracy using minimal measurements on thin section specimens. There is a need further to estimate these physical properties using a simple methodology requiring a minimal number of empirical parameters.
Accordingly, it is a primary object of the present invention to provide an accurate numerical methodology to estimate physical properties of three-dimensional (3D) porous media. It is a further object of the invention to provide such a methodology that operates simply, requiring a minimum of empirical parameters. It is an additional object of the invention to provide such a methodology that utilizes simple and minimal measurements that can be performed easily on specimens of the porous media. It is another object of this invention to provide a methodology for estimating multiple physical properties of the same porous media, thereby permitting cross-relations to be examined.
These objects and advantages are attained by a numerical method for estimating physical properties of three-dimensional (3D) porous media. In accordance with the present invention, this numerical method is based on processed n-ary images of sections of porous media.
In an embodiment of the invention, a thin section is prepared from a specimen of porous media, for example rock, glass, bone, soils, ceramic, sintered granular material, or porous composite material, e.g., concrete. Typically preparation includes filling the pore space portion of the thin section with an epoxy or other polymer resin that has been dyed to contrast optically with the adjacent solid portion. A color micrograph of the thin section is taken and digitized. In some embodiments, a color micrograph is taken of a specimen surface without thinning, with the other preparation steps remaining the same as described above. Using a known digital image processing technique, the micrograph is converted to a binary or higher order n-ary two-dimensional index image. For example, in a binary index image the first binary index can represent the pore space portion and the second binary index can represent the solid portion. In a higher order n-ary index image, various n-ary indices can represent, for example, a pore space portion, a solid, quartz portion, a solid calcite portion, a dolomite portion, a pyrite portion, a siderite portion, a clay portion, a biotite portion, a muscovite portion, and/or a solid feldspar portion respectively. Statistical functions are derived from the two-dimensional n-ary index image, for example, a variogram, a neighborhood template-based multiple point function, an autocorrelation function, and/or a porosity.
Using the statistical functions in cooperation with the n-ary index image, simulated three-dimensional representations of the medium are generated. These representations provide for example a three-dimensional model of porosity in simulated media shaped, for example, as a cube, rectangular prism, cylinder, or other three-dimensional shape having two substantially parallel faces. Each linear dimension of a three-dimensional representation is typically on the order of a predetermined number of autocorrelation lengths. In generating these three-dimensional representations, boundaries can be unconditional or can be advantageously conditioned to the two-dimensional n-ary index image. Typically multiple (e.g., eight or more) equiprobable three-dimensional representations are generated for each n-ary index image, and the values of the physical properties obtained with the multiple representations are averaged to provide a final result.
Desired physical properties can include fluid flow, electrical, and elastic properties, such as permeability, electrical conductivity, and elastic wave velocity. These properties are estimated by performing numerical simulations on the three-dimensional representations. For example, permeability is estimated by using a Lattice-Boltzmann flow simulation as a numerical solver of Navier-Stokes equations for steady flow within a three-dimensional pore space. For electrical conductivity or elastic wave velocity, a finite element numerical solver is advantageously applied. The methodology of the invention can be used to provide estimates of multiple physical properties of the same porous media, permitting cross-relations to be examined.
Thereby the present invention provides an accurate numerical methodology and system to estimate physical properties of three-dimensional (3D) porous media. The invention further provides a relatively simple operating methodology and system requiring a minimum of empirical parameters. Additionally the invention provides a methodology and system that utilize simple and minimal measurements that can be performed easily on specimens of porous media.